Symmetry results for perturbed problems and related questions

Abstract

In this paper we prove a symmetry result for positive solutions of the Dirichlet problem \begin{equation} \begin{cases} -\Delta u=f(u) & \hbox{in }D,\\ u=0 & \hbox{on }\partial D, \end{cases} \tag{0.1} \end{equation} when $f$ satisfies suitable assumptions and $D$ is a small symmetric perturbation of a domain $\Omega$ for which the Gidas-Ni-Nirenberg symmetry theorem applies. We consider both the case when $f$ has subcritical growth and $f(s)=s^{(N+2)/(N-2)}+\lambda s$, $N\ge3$, $\lambda$ suitable positive constant.